There's a key difference in the selection processes between Photoshop and Illustrator.
In Photoshop, everything is a pixel. The background, whether white, black, colored, or transparent are all pixels. So when you create a selection within Photoshop you are selecting those pixels as well as anything which may be visibly different than the background pixels.
In Illustrator, nothing is a pixel. There is no background. Illustrator shows a white page, but if you turn on the transparency grid (
View > Show Transparency Grid) you'll see there is nothing there. Since Illustrator is not pixel-based, the transparent or non-existent background is never part of any selection.
How it's different with the cursor....
In Photoshop the cursor with any of the marquee tools shows you exactly what pixels are being selected. This is because everything is a pixel so everything must be counted in the selection drag.
In Illustrator there are no marquee tools (Lasso Tool doesn't count), so the cursor is merely visually reflecting where you are dragging for reference, but it's never actually selecting the area it's showing as reference. It only selects the objects the cursor touches or which are contained within the drag reference lines. It's not selecting the empty background pixels, because there are no pixels.
The Info Panel in Photoshop shows you the width and height of the selection because all those background pixels matter. The Info Panel in Illustrator shows you the total dimensions of all selected objects because there are no background pixels.
You can not get Illustrator to show you the width and height of a cursor drag primarily because the size of the cursor drag is largely irrelevant and has no actual correlation to the final selection size.
Ultimately it comes down to the core difference between a pixel-based application such as Photoshop and a vector-based application such as Illustrator.
If you need to know the size of a specific area within Illustrator, just draw a standard rectangle around the area and check the dimensions for that, then you can delete the rectangle. And if you're curious, no. There's no way to convert a rectangle to a selected area within Illustrator other than converting it to an artboard or crop marks.